<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">m</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">z1</font> be   <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">m</font>,<a href="margrel1.html#K6">BOOLEAN</a> , <font color="Maroon">z2</font> be   <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">m</font>,<a href="margrel1.html#K6">BOOLEAN</a> ;<br/><b>let </b><font color="Maroon">d1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="margrel1.html#K6">BOOLEAN</a> , <font color="Maroon">d2</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="margrel1.html#K6">BOOLEAN</a> ;<br/>





<b>set </b><font color="Maroon">f</font> =  <a href="margrel1.html#K7">FALSE</a> ;<br/>
<b>set </b><font color="Maroon">t</font> =  <a href="margrel1.html#K8">TRUE</a> ;<br/>
<a NAME="E1:18"/><i><font color="Green">E42</font></i>: 
 <a href="binarith.html#K7">Absval</a> <span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K8">+</a> <font color="Maroon">z2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z1</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <span class="p2">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">siki1</font> =  <a href="binarith.html#K7">Absval</a> <span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K8">+</a> <font color="Maroon">z2</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">siki2</font> =  <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span>;<br/>
<a NAME="E1:18_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K7">Absval</a> <span class="p3">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K8">+</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <span class="p3">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <span class="p2">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="binarith.html#K7">Absval</a> <span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K8">+</a> <font color="Maroon">z2</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E2:18_1"/>
 <a href="binarith.html#K7">Absval</a> <span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K8">+</a> <font color="Maroon">z2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z1</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <span class="p2">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="binarith.html#T47">BINARITH:47</a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green">E43</font></i>:  <a href="binari_2.html#K4" target="_self">Intval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="binarith.html#K8">+</a> <span class="p2">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> = 
 <a href="binari_2.html#K4" target="_self">Intval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K8">+</a> <font color="Maroon">z2</font></span>)</span> <a href="binarith.html#K10">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">d1</font> <a href="binarith.html#K2">'xor'</a> <font color="Maroon">d2</font></span>)</span> <a href="binarith.html#K2">'xor'</a> <span class="p4">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T45">BINARITH:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z1</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p3">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <span class="p3">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d1</font> <a href="binarith.html#K2">'xor'</a> <font color="Maroon">d2</font></span>)</span> <a href="binarith.html#K2">'xor'</a> <span class="p3">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
;<br/>
<i><font color="Green">E44</font></i>:  <a href="binari_2.html#K5" target="_self">Int_add_ovfl</a> <span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <font color="Maroon">d1</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K5">carry</a> <span class="p3">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p4"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p4"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T3">BINARITH:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <font color="Maroon">d1</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K5">carry</a> <span class="p3">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p4"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p4"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T3">BINARITH:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <font color="Maroon">d1</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T44">BINARITH:44</a></i>
;<br/>
<i><font color="Green">E50</font></i>:  <a href="binari_2.html#K6" target="_self">Int_add_udfl</a> <span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">d1</font> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p4"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binarith.html#K5">carry</a> <span class="p4">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T3">BINARITH:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">d1</font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">d2</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binarith.html#K5">carry</a> <span class="p4">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T3">BINARITH:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">d1</font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">d2</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T44">BINARITH:44</a></i>
;<br/>
<a NAME="E5:18"/><i><font color="Green">E51</font></i>: 
 <a href="binari_2.html#K4" target="_self">Intval</a> <span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">d1</font>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:18"/><i><font color="Green">E52</font></i>: 
 <a href="binari_2.html#K4" target="_self">Intval</a> <span class="p1">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">d2</font>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_5"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_5"/>
( ( <font color="Maroon">d1</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  &amp; <font color="Maroon">d2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) or ( <font color="Maroon">d1</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  &amp; <font color="Maroon">d2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) or ( <font color="Maroon">d1</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  &amp; <font color="Maroon">d2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) or ( <font color="Maroon">d1</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  &amp; <font color="Maroon">d2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) )
 </span><b>by </b><i><a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_5_1"><b>suppose </b></a><a NAME="E1:18_5_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">d1</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  &amp; <font color="Maroon">d2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 ;<br/><div class="add"><b>then </b><i><font color="Green">E54</font></i>: <span class="p1">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">d1</font>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z1</font></span>)</span> <a href="binop_2.html#K10">-</a> 0
<b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
<br/>.= 
 <a href="binarith.html#K7">Absval</a> <font color="Maroon">z1</font>

;<br/><i><font color="Green">E59</font></i>: <span class="p1">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">d2</font>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z2</font></span>)</span> <a href="binop_2.html#K10">-</a> 0
<b>by </b><i>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
<br/>.= 
 <a href="binarith.html#K7">Absval</a> <font color="Maroon">z2</font>

;<br/><a NAME="E4:18_5_1"/><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">d1</font> <a href="binarith.html#K2">'xor'</a> <font color="Maroon">d2</font></span>)</span> <a href="binarith.html#K2">'xor'</a> <span class="p1">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font>
 <b>by </b><i/>;<br/><a NAME="E5:18_5_1"/><i><font color="Green">E74</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <font color="Maroon">d1</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font>
 <b>by </b><i/>;<br/><a NAME="E6:18_5_1"/>
<span class="p1">(<span class="default"><font color="Maroon">d1</font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">d2</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i/>;<br/><a NAME="E7:18_5_1"/><b>then </b><i><font color="Green">E75</font></i>: 
 <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">d2</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p3">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><b>thus </b><a NAME="E8:18_5_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_2.html#K4" target="_self">Intval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="binarith.html#K8">+</a> <span class="p4">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <span class="p3">(<span class="default"><a href="binari_2.html#K5" target="_self">Int_add_ovfl</a> <span class="p4">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <span class="p2">(<span class="default"><a href="binari_2.html#K6" target="_self">Int_add_udfl</a> <span class="p3">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p4"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p4"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binari_2.html#K4" target="_self">Intval</a> <span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="binari_2.html#K4" target="_self">Intval</a> <span class="p2">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_5_1_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_5_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_5_1_3_1"/>
(  <a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  or  <a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 </span><b>by </b><i><a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_5_1_3_1_1"><b>suppose </b></a><a NAME="E1:18_5_1_3_1_1"/><i><font color="Green">E76</font></i>: 
 <a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/><div class="add"><a NAME="E2:18_5_1_3_1_1"/><b>then </b><i><font color="Green">E77</font></i>: 
 <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><a NAME="E3:18_5_1_3_1_1"/>
 <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><b>hence </b><a NAME="E4:18_5_1_3_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_2.html#K4" target="_self">Intval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="binarith.html#K8">+</a> <span class="p4">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <span class="p3">(<span class="default"><a href="binari_2.html#K5" target="_self">Int_add_ovfl</a> <span class="p4">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <span class="p2">(<span class="default"><a href="binari_2.html#K6" target="_self">Int_add_udfl</a> <span class="p3">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p4"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p4"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binari_2.html#K4" target="_self">Intval</a> <span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="binari_2.html#K4" target="_self">Intval</a> <span class="p2">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>
 <b>by </b><i>, , , , , , , , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_5_1_3_1_2"><b>suppose </b></a><a NAME="E1:18_5_1_3_1_2"/><i><font color="Green">E78</font></i>: 
 <a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/><div class="add"><a NAME="E2:18_5_1_3_1_2"/><b>then </b>
 <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_2.html#K4" target="_self">Intval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="binarith.html#K8">+</a> <span class="p4">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <span class="p3">(<span class="default"><a href="binari_2.html#K5" target="_self">Int_add_ovfl</a> <span class="p4">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <span class="p2">(<span class="default"><a href="binari_2.html#K6" target="_self">Int_add_udfl</a> <span class="p3">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p4"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p4"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z1</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p3">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, , , , , , , <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z1</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p3">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span> <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z1</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p3">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T32">POWER:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z1</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z2</font></span>)</span>

;<br/><b>hence </b><a NAME="E4:18_5_1_3_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_2.html#K4" target="_self">Intval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="binarith.html#K8">+</a> <span class="p4">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <span class="p3">(<span class="default"><a href="binari_2.html#K5" target="_self">Int_add_ovfl</a> <span class="p4">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <span class="p2">(<span class="default"><a href="binari_2.html#K6" target="_self">Int_add_udfl</a> <span class="p3">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p4"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p4"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binari_2.html#K4" target="_self">Intval</a> <span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="binari_2.html#K4" target="_self">Intval</a> <span class="p2">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_5_2"><b>suppose </b></a><a NAME="E1:18_5_2"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">d1</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  &amp; <font color="Maroon">d2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 ;<br/><div class="add"><a NAME="E2:18_5_2"/><b>then </b><i><font color="Green">E80</font></i>: 
<span class="p1">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">d1</font>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><span class="p1">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">d2</font>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z2</font></span>)</span> <a href="binop_2.html#K10">-</a> 0
<b>by </b><i>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
<br/>.= 
 <a href="binarith.html#K7">Absval</a> <font color="Maroon">z2</font>

;<br/><a NAME="E4:18_5_2"/><b>then </b><i><font color="Green">E81</font></i>: 
<span class="p1">(<span class="default"><a href="binari_2.html#K4" target="_self">Intval</a> <span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="binari_2.html#K4" target="_self">Intval</a> <span class="p2">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z1</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E5:18_5_2"/><i><font color="Green">E82</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">d1</font> <a href="binarith.html#K2">'xor'</a> <font color="Maroon">d2</font></span>)</span> <a href="binarith.html#K2">'xor'</a> <span class="p1">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K9">'not'</a> <span class="p1">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E6:18_5_2"/><i><font color="Green">E83</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <font color="Maroon">d1</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i/>;<br/><a NAME="E7:18_5_2"/><b>then </b><i><font color="Green">E84</font></i>: 
 <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="margrel1.html#K9">'not'</a> <font color="Maroon">d1</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p3">(<span class="default"><a href="margrel1.html#K9">'not'</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><a NAME="E8:18_5_2"/>
<span class="p1">(<span class="default"><font color="Maroon">d1</font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">d2</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i/>;<br/><a NAME="E9:18_5_2"/><b>then </b><i><font color="Green">E85</font></i>: 
 <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">d2</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p3">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><b>thus </b><a NAME="E10:18_5_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_2.html#K4" target="_self">Intval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="binarith.html#K8">+</a> <span class="p4">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <span class="p3">(<span class="default"><a href="binari_2.html#K5" target="_self">Int_add_ovfl</a> <span class="p4">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <span class="p2">(<span class="default"><a href="binari_2.html#K6" target="_self">Int_add_udfl</a> <span class="p3">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p4"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p4"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binari_2.html#K4" target="_self">Intval</a> <span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="binari_2.html#K4" target="_self">Intval</a> <span class="p2">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_5_2_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_5_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_5_2_2_1"/>
(  <a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  or  <a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 </span><b>by </b><i><a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_5_2_2_1_1"><b>suppose </b></a><a NAME="E1:18_5_2_2_1_1"/><i><font color="Green">E86</font></i>: 
 <a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/><div class="add"><a NAME="E2:18_5_2_2_1_1"/><b>then </b><i><font color="Green">E87</font></i>: 
 <a href="margrel1.html#K9">'not'</a> <span class="p1">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/><a NAME="E3:18_5_2_2_1_1"/>
 <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><b>hence </b><a NAME="E4:18_5_2_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_2.html#K4" target="_self">Intval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="binarith.html#K8">+</a> <span class="p4">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <span class="p3">(<span class="default"><a href="binari_2.html#K5" target="_self">Int_add_ovfl</a> <span class="p4">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <span class="p2">(<span class="default"><a href="binari_2.html#K6" target="_self">Int_add_udfl</a> <span class="p3">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p4"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p4"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binari_2.html#K4" target="_self">Intval</a> <span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="binari_2.html#K4" target="_self">Intval</a> <span class="p2">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>
 <b>by </b><i>, , , , , , , , <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_5_2_2_1_2"><b>suppose </b></a><a NAME="E1:18_5_2_2_1_2"/><i><font color="Green">E88</font></i>: 
 <a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/><div class="add"><a NAME="E2:18_5_2_2_1_2"/><b>then </b>
(  <a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="margrel1.html#K7">FALSE</a>  &amp;  <a href="margrel1.html#K9">'not'</a> <span class="p1">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 ;<br/><a NAME="E3:18_5_2_2_1_2"/><b>then </b>
 <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><b>hence </b><a NAME="E4:18_5_2_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_2.html#K4" target="_self">Intval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="binarith.html#K8">+</a> <span class="p4">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <span class="p3">(<span class="default"><a href="binari_2.html#K5" target="_self">Int_add_ovfl</a> <span class="p4">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <span class="p2">(<span class="default"><a href="binari_2.html#K6" target="_self">Int_add_udfl</a> <span class="p3">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p4"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p4"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binari_2.html#K4" target="_self">Intval</a> <span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="binari_2.html#K4" target="_self">Intval</a> <span class="p2">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>
 <b>by </b><i>, , , , , , , <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_5_3"><b>suppose </b></a><a NAME="E1:18_5_3"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">d1</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  &amp; <font color="Maroon">d2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 ;<br/><div class="add"><b>then </b><i><font color="Green">E90</font></i>: <span class="p1">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">d1</font>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z1</font></span>)</span> <a href="binop_2.html#K10">-</a> 0
<b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
<br/>.= 
 <a href="binarith.html#K7">Absval</a> <font color="Maroon">z1</font>

;<br/><a NAME="E3:18_5_3"/><i><font color="Green">E91</font></i>: 
<span class="p1">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">d2</font>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><a NAME="E4:18_5_3"/><i><font color="Green">E92</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">d1</font> <a href="binarith.html#K2">'xor'</a> <font color="Maroon">d2</font></span>)</span> <a href="binarith.html#K2">'xor'</a> <span class="p1">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K9">'not'</a> <span class="p1">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E5:18_5_3"/><i><font color="Green">E93</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <font color="Maroon">d1</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i/>;<br/><a NAME="E6:18_5_3"/><b>then </b><i><font color="Green">E94</font></i>: 
 <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="margrel1.html#K9">'not'</a> <font color="Maroon">d1</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p3">(<span class="default"><a href="margrel1.html#K9">'not'</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><a NAME="E7:18_5_3"/><i><font color="Green">E95</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">d1</font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">d2</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i/>;<br/><a NAME="E8:18_5_3"/><b>then </b><i><font color="Green">E96</font></i>: 
 <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">d2</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p3">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><b>thus </b><a NAME="E9:18_5_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_2.html#K4" target="_self">Intval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="binarith.html#K8">+</a> <span class="p4">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <span class="p3">(<span class="default"><a href="binari_2.html#K5" target="_self">Int_add_ovfl</a> <span class="p4">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <span class="p2">(<span class="default"><a href="binari_2.html#K6" target="_self">Int_add_udfl</a> <span class="p3">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p4"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p4"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binari_2.html#K4" target="_self">Intval</a> <span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="binari_2.html#K4" target="_self">Intval</a> <span class="p2">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_5_3_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_5_3_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_5_3_2_1"/>
(  <a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  or  <a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 </span><b>by </b><i><a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_5_3_2_1_1"><b>suppose </b></a><a NAME="E1:18_5_3_2_1_1"/><i><font color="Green">E97</font></i>: 
 <a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/><div class="add"><a NAME="E2:18_5_3_2_1_1"/><b>then </b><i><font color="Green">E98</font></i>: 
 <a href="margrel1.html#K9">'not'</a> <span class="p1">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/><a NAME="E3:18_5_3_2_1_1"/><i><font color="Green">E99</font></i>: 
 <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><a NAME="E4:18_5_3_2_1_1"/>
 <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><b>hence </b><a NAME="E5:18_5_3_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_2.html#K4" target="_self">Intval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="binarith.html#K8">+</a> <span class="p4">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <span class="p3">(<span class="default"><a href="binari_2.html#K5" target="_self">Int_add_ovfl</a> <span class="p4">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <span class="p2">(<span class="default"><a href="binari_2.html#K6" target="_self">Int_add_udfl</a> <span class="p3">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p4"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p4"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binari_2.html#K4" target="_self">Intval</a> <span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="binari_2.html#K4" target="_self">Intval</a> <span class="p2">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>
 <b>by </b><i>, , , , , , , , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_5_3_2_1_2"><b>suppose </b></a><a NAME="E1:18_5_3_2_1_2"/><i><font color="Green">E100</font></i>: 
 <a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/><div class="add"><a NAME="E2:18_5_3_2_1_2"/><b>then </b><i><font color="Green">E101</font></i>: 
(  <a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="margrel1.html#K7">FALSE</a>  &amp;  <a href="margrel1.html#K9">'not'</a> <span class="p1">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 ;<br/><a NAME="E3:18_5_3_2_1_2"/><i><font color="Green">E102</font></i>: 
 <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><a NAME="E4:18_5_3_2_1_2"/>
 <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><b>hence </b><a NAME="E5:18_5_3_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_2.html#K4" target="_self">Intval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="binarith.html#K8">+</a> <span class="p4">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <span class="p3">(<span class="default"><a href="binari_2.html#K5" target="_self">Int_add_ovfl</a> <span class="p4">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <span class="p2">(<span class="default"><a href="binari_2.html#K6" target="_self">Int_add_udfl</a> <span class="p3">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p4"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p4"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binari_2.html#K4" target="_self">Intval</a> <span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="binari_2.html#K4" target="_self">Intval</a> <span class="p2">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>
 <b>by </b><i>, , , , , , , , , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_5_4"><b>suppose </b></a><a NAME="E1:18_5_4"/><i><font color="Green">E103</font></i>: 
( <font color="Maroon">d1</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  &amp; <font color="Maroon">d2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 ;<br/><div class="add"><a NAME="E2:18_5_4"/><b>then </b><i><font color="Green">E104</font></i>: 
<span class="p1">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">d1</font>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><a NAME="E3:18_5_4"/><i><font color="Green">E105</font></i>: 
<span class="p1">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">d2</font>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><i><font color="Green">E106</font></i>: <span class="p1">(<span class="default"><a href="binari_2.html#K4" target="_self">Intval</a> <span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="binari_2.html#K4" target="_self">Intval</a> <span class="p2">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z1</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z1</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z1</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T32">POWER:32</a></i>
;<br/><a NAME="E5:18_5_4"/><i><font color="Green">E107</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">d1</font> <a href="binarith.html#K2">'xor'</a> <font color="Maroon">d2</font></span>)</span> <a href="binarith.html#K2">'xor'</a> <span class="p1">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font>
 <b>by </b><i/>;<br/><a NAME="E6:18_5_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <font color="Maroon">d1</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i/>;<br/><a NAME="E7:18_5_4"/><b>then </b><i><font color="Green">E108</font></i>: 
 <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="margrel1.html#K9">'not'</a> <font color="Maroon">d1</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p3">(<span class="default"><a href="margrel1.html#K9">'not'</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><a NAME="E8:18_5_4"/><i><font color="Green">E109</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">d1</font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">d2</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K9">'not'</a> <span class="p1">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E9:18_5_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_2.html#K4" target="_self">Intval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="binarith.html#K8">+</a> <span class="p4">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <span class="p3">(<span class="default"><a href="binari_2.html#K5" target="_self">Int_add_ovfl</a> <span class="p4">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <span class="p2">(<span class="default"><a href="binari_2.html#K6" target="_self">Int_add_udfl</a> <span class="p3">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p4"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p4"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binari_2.html#K4" target="_self">Intval</a> <span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="binari_2.html#K4" target="_self">Intval</a> <span class="p2">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_5_4_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_5_4_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_5_4_2_1"/>
(  <a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  or  <a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 </span><b>by </b><i><a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_5_4_2_1_1"><b>suppose </b></a><a NAME="E1:18_5_4_2_1_1"/><i><font color="Green">E110</font></i>: 
 <a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/><div class="add"><a NAME="E2:18_5_4_2_1_1"/><b>then </b><i><font color="Green">E111</font></i>: 
 <a href="margrel1.html#K9">'not'</a> <span class="p1">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/><a NAME="E3:18_5_4_2_1_1"/>
 <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><b>hence </b><a NAME="E4:18_5_4_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_2.html#K4" target="_self">Intval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="binarith.html#K8">+</a> <span class="p4">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <span class="p3">(<span class="default"><a href="binari_2.html#K5" target="_self">Int_add_ovfl</a> <span class="p4">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <span class="p2">(<span class="default"><a href="binari_2.html#K6" target="_self">Int_add_udfl</a> <span class="p3">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p4"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p4"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binari_2.html#K4" target="_self">Intval</a> <span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="binari_2.html#K4" target="_self">Intval</a> <span class="p2">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>
 <b>by </b><i>, , , , , , , , <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_5_4_2_1_2"><b>suppose </b></a><a NAME="E1:18_5_4_2_1_2"/><i><font color="Green">E112</font></i>: 
 <a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/><div class="add"><a NAME="E2:18_5_4_2_1_2"/><b>then </b><i><font color="Green">E113</font></i>: 
(  <a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="margrel1.html#K7">FALSE</a>  &amp;  <a href="margrel1.html#K9">'not'</a> <span class="p1">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 ;<br/><a NAME="E3:18_5_4_2_1_2"/><i><font color="Green">E114</font></i>: 
 <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><a NAME="E4:18_5_4_2_1_2"/>
 <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_2.html#K4" target="_self">Intval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="binarith.html#K8">+</a> <span class="p4">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <span class="p3">(<span class="default"><a href="binari_2.html#K5" target="_self">Int_add_ovfl</a> <span class="p4">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <span class="p2">(<span class="default"><a href="binari_2.html#K6" target="_self">Int_add_udfl</a> <span class="p3">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p4"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p4"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z1</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i>, , , , , , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z1</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z1</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T32">POWER:32</a></i>
;<br/><b>hence </b><a NAME="E6:18_5_4_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_2.html#K4" target="_self">Intval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="binarith.html#K8">+</a> <span class="p4">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <span class="p3">(<span class="default"><a href="binari_2.html#K5" target="_self">Int_add_ovfl</a> <span class="p4">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <span class="p2">(<span class="default"><a href="binari_2.html#K6" target="_self">Int_add_udfl</a> <span class="p3">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p4"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p4"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binari_2.html#K4" target="_self">Intval</a> <span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="binari_2.html#K4" target="_self">Intval</a> <span class="p2">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div>
